چکیده انگلیسی

In this paper we analyze the optimal joint decisions of when, how and how much to replenish customers with products of varying ages. We discuss the main features of the problem arising in the joint replenishment and delivery of perishable products, and we model them under general assumptions. We then solve the problem by means of an exact branch-and-cut algorithm, and we test its performance on a set of randomly generated instances. Our algorithm is capable of computing optimal solutions for instances with up to 30 customers, three periods, and a maximum age of two periods for the perishable product. For the unsolved instances the optimality gap is always small, less than 1.5% on average for instances with up to 50 customers. We also implement and compare two suboptimal selling priority policies with an optimized policy: always sell the oldest available items first to avoid spoilage, and always sell the fresher items first to increase revenue.

مقدمه انگلیسی

Inventory control constitutes an important logistics operation, especially when products have a limited shelf life. Keeping the right inventory levels guarantees that the demand is satisfied without incurring unnecessary holding or spoilage costs. Several inventory control models are available [3], many of which include a specific treatment of perishable products [30].
Problems related to the management of perishable products׳ inventories arise in several areas. Applications of inventory control of perishable products include blood management and distribution [5], [9], [17], [18], [20], [25], [26] and [33], as well as the handling of radioactive and chemical materials [1], [11] and [37], of food such as dairy products, fruits and vegetables [4], [12], [29], [31], [35] and [36], and of fashion apparel [28]. Several inventory management models have been specifically derived for perishable items, such as the periodic review with minimum and maximum order quantity of Haijema [15], and the periodic review with service level considerations of Minner and Transchel [24]. Reviews of the main models and algorithms in this area can be found in Nahmias [30] and in Karaesmen et al. [19]. A unified analytical approach to the management of supply chain networks for time-sensitive products is provided in Nagurney et al. [27].
Efficient delivery planning can provide further savings in logistics operations. The optimization of vehicle routes is one of the most developed fields in operations research [21]. The integration of inventory control and vehicle routing yields a complex optimization problem called inventory-routing whose aim is to minimize the overall costs related to vehicle routes and inventory control. Recent overviews of the inventory-routing problem (IRP) are those of Andersson et al. [2] and of Coelho et al. [8].
The joint inventory management and distribution of perishable products, which is the topic of this paper, gives rise to the perishable inventory-routing problem (PIRP). Nagurney and Masoumi [25] and Nagurney et al. [26] studied the distribution and relocation of human blood in a stochastic demand context, considering the perishability and waste of blood related to age and to the limited capacity of blood banks. Hemmelmayr et al. [16] studied the case of blood inventory control with predetermined fixed routes and stochastic demand. The problem was solved heuristically by integer programming and variable neighborhood search. Gumasta et al. [14] incorporated transportation issues in an inventory control model restricted to two customers only. Custódio and Oliveira [10] proposed a strategical heuristic analysis of the distribution and inventory control of several frozen groceries with stochastic demand. Mercer and Tao [23] studied the weekly food distribution problem of a supermarket chain, without considering product age. A theoretical paper developing a column generation approach was presented by Le et al. [22] to provide solutions to a PIRP. The optimality gap was typically below 10% for instances with eight customers and five periods under the assumptions of fixed shelf life and flat value throughout the life of the product.
This paper makes several scientific contributions. We first classify and discuss the main assumptions underlying the management of perishable products. We then formulate the PIRP as a mixed integer linear program (MILP) for the most general case, and we also model it to handle the cases where retailers always sell older items first, and where they sell fresher items first. We devise an exact branch-and-cut algorithm for the solution of the various models. To the best of our knowledge, this is the first time an IRP is modeled and solved exactly under general assumptions in the context of perishable products management. Our models do not require any assumption on the shape of the product revenue and inventory cost functions. We also establish some relationships between the PIRP and the multi-product IRP recently studied by the authors [7].
The remainder of the paper is organized as follows. In Section 2 we provide a formal description of the PIRP. In Section 3 we present our MILP model and its two variants just described, including new valid inequalities. This is followed by a description of the branch-and-cut algorithm in Section 4. Computational experiments are presented in Section 5. Section 6 concludes the paper.

نتیجه گیری انگلیسی

We have introduced the joint replenishment and inventory control of perishable products. We have modeled the problem under general assumptions as a MILP, and we have solved it exactly by branch-and-cut. We have also introduced, modeled and solved exactly two variants of the problem defined by applying the OF and the FF selling priority policies, in which the retailer sells with higher priority older and fresher items, respectively. Our model remains linear even when the product revenue decreases in a non-linear or even in a non-convex fashion over time. It keeps track of the number of items of each age, and considers different holding costs for products of different ages. The model identifies products of different ages independently from each other, which is very similar to dealing with several products, as in a multi-product environment, but not identical since the state of the product changes over time. The model optimally determines which items to sell at each period based on the trade-off between cost and revenue. The algorithm can effectively compute optimal joint replenishment and delivery decisions for perishable products in an inventory-routing context for medium size instances. We have also shown that on our testbed, the profit changes drastically depending on the shape of the revenue of the product. On monotonically decreasing revenue functions, the value of the solution obtained under the OP is reduced when an OF policy is applied, but the decrease is only marginal under an FF policy. Extensive computational experiments carried out on randomly generated instances support these conclusions.